The butterfly theorem is a classical result in

Euclidean geometry Euclidean geometry is a mathematical system attributed to ancient Greek mathematics, Greek mathematician Euclid, which he described in his textbook on geometry, ''Euclid's Elements, Elements''. Euclid's approach consists in assuming a small set ...

, which can be stated as follows:Johnson, Roger A., ''Advanced Euclidean Geometry'', Dover Publ., 2007 (orig. 1929). Let be the

midpoint In geometry, the midpoint is the middle point of a line segment. It is equidistant from both endpoints, and it is the centroid both of the segment and of the endpoints. It bisects the segment. Formula The midpoint of a segment in ''n''-dim ...

of a chord of a

circle A circle is a shape consisting of all point (geometry), points in a plane (mathematics), plane that are at a given distance from a given point, the Centre (geometry), centre. The distance between any point of the circle and the centre is cal ...

, through which two other chords and are drawn; and intersect chord at and correspondingly. Then is the midpoint of .

Proof

A formal proof of the theorem is as follows: Let the perpendiculars and be dropped from the point on the straight lines and respectively. Similarly, let and be dropped from the point perpendicular to the straight lines and respectively. Since ::

\triangle MXX' \sim \triangle MYY',

= ,

\triangle MXX'' \sim \triangle MYY'',

= ,

\triangle AXX' \sim \triangle CYY'',

= ,

\triangle DXX'' \sim \triangle BYY',

= .

From the preceding equations and the intersecting chords theorem, it can be seen that :

\left(\right)^2 =  ,

= ,

= ,

= ,

= ,

since . So, :

= .

Cross-multiplying in the latter equation, :

=  .

Cancelling the common term :

from both sides of the equation yields :

= ,

hence , since MX, MY, and PM are all positive, real numbers. Thus, is the midpoint of . Other proofs exist, including one using

projective geometry In mathematics, projective geometry is the study of geometric properties that are invariant with respect to projective transformations. This means that, compared to elementary Euclidean geometry, projective geometry has a different setting (''p ...

History

Proving the butterfly theorem was posed as a problem by William Wallace in ''The Gentleman's Mathematical Companion'' (1803). Three solutions were published in 1804, and in 1805 Sir William Herschel posed the question again in a letter to Wallace. Reverend Thomas Scurr asked the same question again in 1814 in the ''Gentleman's Diary or Mathematical Repository''.William Wallace's 1803 Statement of the Butterfly Theorem

cut-the-knot Alexander Bogomolny (January 4, 1948 July 7, 2018) was a Soviet Union, Soviet-born Israeli Americans, Israeli-American mathematician. He was Professor Emeritus of Mathematics at the University of Iowa, and formerly research fellow at the Moscow ...

, retrieved 2015-05-07.

References

External links

The Butterfly Theorem
at

A Better Butterfly Theorem
at

Proof of Butterfly Theorem
at

PlanetMath PlanetMath is a free content, free, collaborative, mathematics online encyclopedia. Intended to be comprehensive, the project is currently hosted by the University of Waterloo. The site is owned by a US-based nonprofit corporation, "PlanetMath.org ...

The Butterfly Theorem
by Jay Warendorff, the

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. * {{MathWorld , title=Butterfly Theorem , urlname=ButterflyTheorem Euclidean plane geometry Theorems about circles Articles containing proofs